Re: help ! if R is regular ring, e and f are idempotents elements then Re+Rf=R(e+f)

Notice if R is regular, then such that and so Now how do we know that x isn't a zero divisor? and thus ? Well, if then and so which means , which cannot be so, and thus, , so , which basically means that every non zero element of R is a unit since we picked x at random from R. So, since there exist inverses for f and e, and , so basically both

Re: help ! if R is regular ring, e and f are idempotents elements then Re+Rf=R(e+f)

because i wanted the result that so which would mean x was a unit. Now if you remember, if is a zero divisor then a element such that but neither a nor b is 0. So if then i cannot get the result i want, which is that x is a unit

Re: help ! if R is regular ring, e and f are idempotents elements then Re+Rf=R(e+f)

Originally Posted by jakncoke

Notice if R is regular, then such that and so Now how do we know that x isn't a zero divisor? and thus ? Well, if then and so which means , which cannot be so, and thus, , so , which basically means that every non zero element of R is a unit since we picked x at random from R. So, since there exist inverses for f and e, and , so basically both

we cannot conclude from yx - 1 ≠ 0 that x(yx - 1) ≠ 0. if this were so, every von neumann regular ring would be a field, and this is not so.

here is a counter-example:

let R = M(2,Q), the ring of 2x2 matrices over the rationals.

if a 2x2 rational matrix x is invertible, we may take y = x-1.

so suppose x is singular. some easy cases, first off:

if x = 0, any y will do.

if x =

, we may take y =

.

otherwise we have x = PAQ, where A is the matrix:

and P,Q are invertible matrices (which we get by keeping track of which elementary row/column operations reduce x to A).

if we label the matrix:

,

and set y = Q-1BP-1, we have:

x = PAQ = P(ABA)Q (from above, A = ABA)

= PA(QQ-1)B(P-1P)AQ

= (PAQ)(Q-1BP-1)(PAQ) = xyx.

so this ring is indeed (von neumann) regular, and we have found a non-unit in it (which is also a zero-divisor).

*********

i haven't had time to look at this properly, but the fact that e and f are idempotent must be important.